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Spatial decay of discretely self-similar solutions to the Navier–Stokes equations

Zachary Bradshaw and Patrick Phelps

Vol. 5 (2023), No. 2, 377–407

Forward self-similar and discretely self-similar weak solutions of the Navier–Stokes equations are known to exist globally in time for large self-similar and discretely self-similar initial data and are known to be regular outside of a space-time paraboloid. We establish spatial decay rates for such solutions which hold in the region of regularity provided the initial data has locally subcritical regularity away from the origin. In particular, we lower the Hölder regularity of the data required to obtain an optimal decay rate for the nonlinear part of the flow compared to the existing literature, establish new decay rates without logarithmic corrections for some smooth data, provide new decay rates for solutions with rough data, and, as an application of our decay rates, provide new upper bounds on how rapidly potentially nonunique, discretely self-similar local energy solutions can separate away from the origin.

Navier–Stokes, asymptotic expansion, decay, self-similar
Mathematical Subject Classification
Primary: 35Q30, 76D05
Received: 16 February 2022
Accepted: 3 January 2023
Published: 26 June 2023
Zachary Bradshaw
Department of Mathematical Sciences
University of Arkansas
Fayetteville, AR
United States
Patrick Phelps
Department of Mathematical Sciences
University of Arkansas
Fayetteville, AR
United States